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Abstract—This paper discusses the construction of polynomial
and non-polynomial splines of the fourth order of approximation.
The behavior of the Lebesgue constants for the left, the right, and
the middle continuous cubic polynomial splines are considered.
The non-polynomial splines are used for the construction of the
special central difference approximation. The approximation of
functions, and the solving of the boundary problem with the
polynomial and non-polynomial splines are discussed. Numerical
examples are done.
Keywords— Polynomial splines, nonpolynomial splines,
boundary value problem.
I. INTRODUCTION
ECENYLY, polynomial and non-polynomial splines are
often used to solve a variety of problems: in
approximating functions, in constructing numerical schemes
for solving boundary value problems, and for solving integral
equations (see [1]-[5]). Still a lot of attention is paid to the
construction and use of suitable splines in the Galerkin method
[6]. At the same time, in the construction and application of
splines, little attention is paid to the Lebesgue functions and the
Lebesgue constants. The Lebesgue constants are named after
Henri Lebesgue. The Lebesgue constants give an idea of how
good the approximation of functions are in comparison with the
best polynomial approximation of the function (see [7]-[8]).
When constructing interpolation and when studying the
quality of the approximation, the authors often use the constant
Lebesgue (see [9]-[12]). The exact values of integral Lebesgue
constants of generalized special local splines are calculated in
[9]. Using the analysis of interpolation error [10], facilitates
better node placement for minimizing interpolation error
compared to the traditional approach of minimizing the
Lebesgue constant as a proxy for interpolation error. The exact
values for the uniform Lebesgue constants of interpolating L-
splines are found in [11]. The L-splines, which are bound on the
real axis, have equidistant knots and correspond to the linear
third-order differential operator. It is proved in [12] that the
uniform Lebesgue constant (the norm of a linear operator from
C to C) of local cubic splines with equally spaced nodes, which
preserve cubic polynomials, is equal to 11/9.
A spline defect is usually called the difference between the
degree of spline and smoothness. Polynomial cubic splines of
the maximum defect are used in solving many problems of
mathematical physics. Prof. S.G.Mikhlin and prof.
Yu.K.Demjanovich have devoted lot of attention to the
construction and study of polynomial local splines (see, for
example [13]-[14]). In paper [13], methods are given for
constructing splines of generalized smoothness in the case of
splines of the Lagrangian type on a differentiable manifold. In
paper [14], polynomial splines of the Hermitian type are
constructed on a uniform grid of nodes.
Earlier, the Lebesgue constant was discussed in connection
with the construction of the splines of the maximal defect on a
finite grid of nodes in paper [15]. It should be noted that the
method of construction of the polynomial coordinate functions
proposed in paper [14], as well as the methods of construction
of variational-difference equation uses these splines.
Trigonometric splines [16] are often used in solving various
problems as they help construct a better result than when we
use the polynomial splines.
In this paper we will first dwell briefly on the construction and
properties of the polynomial and non-polynomial splines of the
fourth order approximation. These non-polynomial splines
have the properties of both polynomial and trigonometric
splines. Next, we introduce the concepts of functions and
Lebesgue constants. Then we consider the use of splines to
solve the boundary value problem by the difference method. To
construct an algorithm for solving the difference method, we
need formulas for the approximate calculation of the first and
the second derivatives of the function. These formulas will be
constructed on the basis of the non-polynomial splines obtained
in this paper. These formulas can be applied to solve boundary
value problems. Note that recently the authors have been trying
to find alternative approaches for solving differential equations.
In paper [18], the numerical solution of nonlinear two-
dimensional parabolic partial differential equations with initial
and Dirichlet boundary conditions is considered. The time
Continuous local splines of the fourth order of
approximation and boundary value problem
I.G.Burova
St. Petersburg State University
7/9 Universitetskaya nab., St.Petersburg, 199034 Russia
Russia, burovaig@mail.ru, i.g.burova@spbu.ru
Received: June 11, 2020, Revised: July 30, 2020. Accepted: August 7, 2020. Published: August 13, 2020.
R
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 440
derivative is approximated using a finite difference scheme
whereas space derivatives are approximated using the Haar
wavelet collocation method.
II. THE POLYNOMIAL CUBIC SPLINES CONSTRUCTION AND LEBESGUE
FUNCTIONS
Let 𝑙, 𝑠, 𝑚, 𝑁 be non-negative integers such that 𝑙 ≥ 1, 𝑠 ≥ 1, 𝑙 + 𝑠 = 𝑚 + 1, 𝑁 ≥ 4. Let the function 𝑢(𝑥) be such that 𝑢 ∈𝐶𝑚+1 ([𝑎, 𝑏]), 𝑎, 𝑏 are real. The nodes 𝑥𝑖 ∈ [𝑎, 𝑏], 𝑖 = 0, … , 𝑁, such that 𝑎 = 𝑥0 < ⋯ < 𝑥𝑖−1 < 𝑥𝑖 < 𝑥𝑖+1 < … < 𝑥𝑁 = 𝑏.
The approximation 𝑈(𝑥) of function 𝑢(𝑥) in the interval
[𝑥𝑗 , 𝑥𝑗+1] will be constructed in the form
𝑈(𝑥) = ∑ 𝑢(𝑥𝑖)𝑤𝑖(𝑥),
𝑖
(1)
where 𝑤𝑖(𝑥) are the interpolation basis polynomial splines
which are determined from the conditions:
∑ 𝜑α(𝑥𝑖)𝑤𝑖(𝑥) = 𝜑α(𝑥), α = 1, … , 𝑚 + 1.
𝑖
The system of functions 𝜑α is the Chebyshev system. Here we
assume that 𝑠𝑢𝑝𝑝 𝑤𝑖 = [𝑥𝑖−𝑠, 𝑥𝑖+𝑙 ]. Assuming that the
multiplicity of the covering of an arbitrary point 𝑥 by the
supports of basis splines 𝑤𝑖(𝑥) is almost everywhere equal to
𝑚 + 1. It is easy to see that in sum (1) for 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1] the
amount of terms is small:
𝑈(𝑥) = ∑ 𝑢(𝑥𝑖)
𝑗+𝑠
𝑖=𝑗−𝑙+1
𝑤𝑖(𝑥).
In this case the basic splines 𝑤𝑖(𝑥) are determined from the
system of equations uniquely:
∑ 𝜑α(𝑥𝑖)
𝑗+𝑠
𝑖=𝑗−𝑙+1
𝑤𝑖(𝑥) = 𝜑α(𝑥), α = 1, 2, … , 𝑚 + 1. (2)
Further, we assume that 𝜑1 = 1. With this assumption, the
first equation in system (2) has the form:
∑ 𝑤𝑖(𝑥)
𝑗+𝑠
𝑖=𝑗−𝑙+1
= 1.
The splines 𝑤𝑖(𝑥) which are constructed in this way have
the following properties:
𝑤𝑖(𝑥𝑖) = 1, 𝑤𝑖(𝑥𝑗) = 0, for 𝑗 ≠i.
𝑢(𝑥) − 𝑈(𝑥) = 0 for 𝑢(𝑥) = 𝜑𝛼(𝑥), 𝛼 = 1, 2, … , 𝑚 + 1.
𝑤𝑗 ∈ 𝐶([𝑥𝑗−𝑠, 𝑥𝑗+𝑙 ]).
Near the boundaries of the finite interval [𝑎, 𝑏], we can use
either left splines, or right splines. Moreover, to construct an
approximation we will only use function values at the grid
nodes belonging to the interval [𝑎, 𝑏]. As is known, in the case of classical interpolation
polynomials, the Lebesgue function and the Lebesgue constant
are related to the approximation of a function and the
convergence of a sequence of interpolation polynomials to a
function.
The error estimation arising due to inaccurate assignment of
the function value in the grid nodes can also be investigated
using Lebesgue functions.
The Lebesgue function of the basic splines 𝑤𝑖(𝑥) with
respect to the interval 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1] will be called
𝜆𝑗,𝑚+1(𝑥) = ∑ |𝑤𝑖(𝑥)|,
𝑗+𝑠
𝑖=𝑗−𝑙+1
𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1]. (3)
The Lebesgue constant of the basic splines 𝑤𝑖(𝑥) with
respect to the interval [𝑥𝑗 , 𝑥𝑗+1] will be called the quantity
𝜆𝑗,𝑚+1(𝑥) = max𝑥∈[𝑥𝑗,𝑥𝑗+1]
𝜆𝑗,𝑚+1(𝑥).
From equation (3) we obtain the inequality 𝜆𝑗,𝑚+1(𝑥) ≥ 1,
𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1].
Let 𝐹 be a real linear normed space of real functions 𝑓(𝑥) on
the interval [𝑎, 𝑏], 𝑛 = 𝑚 + 1. We approximate functions from
𝐹 by generalized polynomials of the form
∑ 𝑐α𝜑α(𝑥).
𝑛
α=1
Here 𝑐𝑘 are real numbers.
Let 𝐹𝑛 = 𝑠𝑝𝑎𝑛 {𝜑1, … , 𝜑𝑛}, where 𝜑1, … , 𝜑𝑛 are the basis
functions in 𝐹𝑛. Consider the function
𝑔(𝑐1, … , 𝑐𝑛) = ‖𝑢 − ∑ 𝑐α𝜑α
𝑛
α=1
‖.
The best approximation is the number
𝑑 = 𝐸𝐹𝑛(𝑢) = inf
𝑐1,…,𝑐𝑛∈ℝ𝑛𝑔(𝑐1, … , 𝑐𝑛). Suppose 𝑑 = ‖𝑢 − 𝜓‖.
In this case 𝜓 is called the element of the best approximation.
In this paper we consider the functions 𝜑α(𝑥) = 𝑥α−1, α =1, 2, 3, 4.
Lemma 1. Let function 𝑢 be such that 𝑢 ∈ 𝐶[𝑎, 𝑏]. The
following inequalities are valid:
|𝑈| ≤ 𝜆𝑗,𝑚+1 (𝑥)‖𝑢‖𝐶[𝑥𝑗−𝑙+1,𝑥𝑗+𝑠+1],
|𝑈| ≤ 𝜆𝑗,𝑚+1 ‖𝑢‖𝐶[𝑥𝑗−𝑙+1,𝑥𝑗+𝑠+1].
Proof. The first inequality follows from the relation
𝑈(𝑥) = ∑ 𝑢(𝑥𝑘)
𝑗+𝑠
𝑘=𝑗−𝑙+1
𝑤𝑘(𝑥).
Calculating the maximum on the right side of this expression
when 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], we get the second relation.
The proof is complete.
Here we discuss similar problems in the case of
approximation by polynomial cubic splines. To construct the
right, left and middle continuous cubic splines, we need to take
𝑚 = 3 and take the parameter 𝑙, respectively, equal to 1, 2 or 3.
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 441
In the case 𝑙 = 1, in the approximation 𝑈, we use the values of
the function 𝑢 lying to the right of 𝑎.Thus, we use only grid
nodes that only belong to the finite interval [𝑎, 𝑏]. In the case
𝑙 = 3, in the approximation 𝑈, we use the values of the function
𝑢 lying to the left of 𝑏.Thus, we use only grid nodes that belong
only to the finite interval [𝑎, 𝑏], In the case of 𝑙 = 2, the
approximation error turns out to be smaller than in the case of
𝑙 = 1 or 𝑙 = 3. In this case, we need to ensure that in the
approximation only grid nodes that are belong to the interval
[𝑎, 𝑏] are used.
The middle cubic splines give a smaller constant in the
approximation error. In the case of a finite interval [𝑎, 𝑏], they
can be used at intervals spaced from the boundaries at a distance
not less than two grid intervals, otherwise, we need the values
of the function at the grid nodes that are outside the interval
[𝑎, 𝑏].
We will use the following notations: 𝑤𝑗𝑀(𝑥), 𝑤𝑗
𝑅(𝑥), 𝑤𝑗𝐿(𝑥)
are the middle, the right and the left basis cubic polynomial
splines, and 𝜆4𝑀, 𝜆4
𝑅 , 𝜆4𝐿 are the Lebesgue constants
corresponding to them (we omit the index 𝑗).
The approximation with the right basis splines can be written
as follows:
𝑈𝑅(𝑥) = 𝑢(𝑥𝑗)𝑔𝑗 + 𝑢(𝑥𝑗+1)𝑔𝑗+1 + 𝑢(𝑥𝑗+2)𝑔𝑗+2
+𝑢(𝑥𝑗+3)𝑔𝑗+3,
where
𝑔𝑗 =(𝑥 − 𝑥𝑗+1)(𝑥 − 𝑥𝑗+2)(𝑥 − 𝑥𝑗+3)
(𝑥𝑗 − 𝑥𝑗+1)(𝑥𝑗 − 𝑥𝑗+2)(𝑥𝑗 − 𝑥𝑗+3),
𝑔𝑗+1 =(𝑥 − 𝑥𝑗)(𝑥 − 𝑥𝑗+2)(𝑥 − 𝑥𝑗+3)
(𝑥𝑗+1 − 𝑥𝑗)(𝑥𝑗+1 − 𝑥𝑗+2)(𝑥𝑗+1 − 𝑥𝑗+3),
𝑔𝑗+2 =(𝑥 − 𝑥𝑗)(𝑥 − 𝑥𝑗+1)(𝑥 − 𝑥𝑗+3)
(𝑥𝑗+2 − 𝑥𝑗)(𝑥𝑗+2 − 𝑥𝑗+1)(𝑥𝑗+2 − 𝑥𝑗+3),
𝑔𝑗+3 =(𝑥 − 𝑥𝑗)(𝑥 − 𝑥𝑗+2)(𝑥 − 𝑥𝑗+2)
(𝑥𝑗+3 − 𝑥𝑗)(𝑥𝑗+3 − 𝑥𝑗+2)(𝑥𝑗+3 − 𝑥𝑗+2).
Lemma 2. Let 𝑁 ≥ 4, 𝑚 = 3 and 𝑇(𝑥), 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], be the
best polynomial approximation of the function 𝑢(𝑥), 𝑢 ∊
𝐶([𝑥𝑗 , 𝑥𝑗+3]). Let 𝐸(𝑢) be the best approximation of the
function 𝑢(𝑥) with 𝑇(𝑥). When approximating the function
𝑢(𝑥), 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1] with the right basis splines, the inequality
is valid:
|𝑈𝑅(𝑥) − 𝑢(𝑥)| ≤ (1 + 𝜆4𝑅)𝐸(𝑢).
Proof. We use the relation
𝑈𝑅(𝑥) = ∑ 𝑢(𝑥𝑖)𝑤𝑖𝑅
𝑗+3
𝑖=𝑗
(𝑥)
for the approximation 𝑢(𝑥) when 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1]. Obviously, the
relation is valid:
𝑇(𝑥) = ∑ 𝑇(𝑥𝑖)𝑤𝑖𝑅
𝑗+3
𝑖=𝑗
(𝑥).
Now we have
|𝑈𝑅(𝑥) − 𝑢(𝑥)| = |𝑈𝑅(𝑥) − 𝑇(𝑥)|
+|𝑇(𝑥) − 𝑢(𝑥)| ≤ 𝐸(𝑢) + |∑(𝑇(𝑥𝑖)−𝑢(𝑥𝑖))𝑤𝑖𝑅
𝑗+3
𝑖=𝑗
(𝑥)|
≤ 𝐸(𝑢)(1 + 𝜆4𝑅).
The proof is complete.
The approximation with the left basis splines can be written
as follows:
𝑈𝐿(𝑥) = 𝑢(𝑥𝑗−2)𝑔𝑗−2 + 𝑢(𝑥𝑗−1)𝑔𝑗−1 + 𝑢(𝑥𝑗)𝑔𝑗
+𝑢(𝑥𝑗+1)𝑔𝑗+1,
where
𝑔𝑗−2 =(𝑥 − 𝑥𝑗−1)(𝑥 − 𝑥𝑗)(𝑥 − 𝑥𝑗+1)
(𝑥𝑗−2 − 𝑥𝑗−1)(𝑥𝑗−2 − 𝑥𝑗)(𝑥𝑗−2 − 𝑥𝑗+1),
𝑔𝑗−1 =(𝑥 − 𝑥𝑗−2)(𝑥 − 𝑥𝑗)(𝑥 − 𝑥𝑗+1)
(𝑥𝑗−1 − 𝑥𝑗−2)(𝑥𝑗−1 − 𝑥𝑗)(𝑥𝑗−1 − 𝑥𝑗+1),
𝑔𝑗 =(𝑥 − 𝑥𝑗−2)(𝑥 − 𝑥𝑗−1)(𝑥 − 𝑥𝑗+1)
(𝑥𝑗 − 𝑥𝑗−2)(𝑥𝑗 − 𝑥𝑗−1)(𝑥𝑗 − 𝑥𝑗+1),
𝑔𝑗+1 =(𝑥 − 𝑥𝑗−2)(𝑥 − 𝑥𝑗−1)(𝑥 − 𝑥𝑗−2)
(𝑥𝑗+1 − 𝑥𝑗−2)(𝑥𝑗+1 − 𝑥𝑗−1)(𝑥𝑗+1 − 𝑥𝑗−2).
Lemma 3. Let 𝑁 ≥ 4, 𝑚 = 3 and 𝑇(𝑥), 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], be the
best polynomial approximation of the function 𝑢(𝑥), 𝑢 ∊
𝐶([𝑥𝑗−2, 𝑥𝑗+1]). Let 𝐸(𝑢) be the best approximation of the
function 𝑢(𝑥) with 𝑇(𝑥).
When approximating the function 𝑢(𝑥), 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1] with
the left basis splines, the inequality is valid:
|𝑈𝐿(𝑥) − 𝑢(𝑥)| ≤ (1 + 𝜆4𝐿)𝐸(𝑢).
Proof. We use the relation
𝑈𝐿(𝑥) = ∑ 𝑢(𝑥𝑖)𝑤𝑖𝐿
𝑗+1
𝑖=𝑗−2
(𝑥)
for the approximation 𝑢(𝑥) when 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1].
Obviously, the relation is valid:
𝑇(𝑥) = ∑ 𝑇(𝑥𝑖)𝑤𝑖𝐿
𝑗+1
𝑖=𝑗−2
(𝑥).
Now we have
|𝑈𝐿(𝑥) − 𝑢(𝑥)| = |𝑈𝐿(𝑥) − 𝑇(𝑥)| +
|𝑇(𝑥) − 𝑢(𝑥)| ≤ 𝐸(𝑢) + | ∑ (𝑇(𝑥𝑖)−𝑢(𝑥𝑖))𝑤𝑖𝐿
𝑗+1
𝑖=𝑗−2
(𝑥)|
≤ 𝐸(𝑢)(1 + 𝜆4𝐿).
The proof is complete.
The approximation with the middle basis splines can be
written as follows:
𝑈𝑀(𝑥) = 𝑢(𝑥𝑗−1)𝑔𝑗−1 + 𝑢(𝑥𝑗)𝑔𝑗 + 𝑢(𝑥𝑗+1)𝑔𝑗+1
+𝑢(𝑥𝑗+2)𝑔𝑗+2,
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 442
where
𝑔𝑗−1 =(𝑥 − 𝑥𝑗)(𝑥 − 𝑥𝑗+1)(𝑥 − 𝑥𝑗+2)
(𝑥𝑗−1 − 𝑥𝑗)(𝑥𝑗−1 − 𝑥𝑗+1)(𝑥𝑗−1 − 𝑥𝑗+2),
𝑔𝑗 =(𝑥 − 𝑥𝑗−1)(𝑥 − 𝑥𝑗+1)(𝑥 − 𝑥𝑗+2)
(𝑥𝑗 − 𝑥𝑗−1)(𝑥𝑗 − 𝑥𝑗+1)(𝑥𝑗 − 𝑥𝑗+2),
𝑔𝑗+1 =(𝑥 − 𝑥𝑗−1)(𝑥 − 𝑥𝑗)(𝑥 − 𝑥𝑗+2)
(𝑥𝑗+1 − 𝑥𝑗−1)(𝑥𝑗+1 − 𝑥𝑗)(𝑥𝑗+1 − 𝑥𝑗+2),
𝑔𝑗+2 =(𝑥 − 𝑥𝑗−1)(𝑥 − 𝑥𝑗+1)(𝑥 − 𝑥𝑗)
(𝑥𝑗+2 − 𝑥𝑗−1)(𝑥𝑗+2 − 𝑥𝑗+1)(𝑥𝑗+2 − 𝑥𝑗).
Lemma 4. Let 𝑁 ≥ 4, 𝑚 = 3 and 𝑇(𝑥), 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], be the
best polynomial approximation of the function 𝑢(𝑥), 𝑢 ∊
𝐶([𝑥𝑗−1, 𝑥𝑗+2]). Let 𝐸(𝑢) be the best approximation of the
function 𝑢(𝑥) with 𝑇(𝑥).
When approximating the function 𝑢(𝑥), 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1] with
the middle basis splines, the inequality is valid:
|𝑈𝑀(𝑥) − 𝑢(𝑥)| ≤ (1 + 𝜆4𝑀)𝐸(𝑢).
Proof. We use the relation
𝑈𝑀(𝑥) = ∑ 𝑢(𝑥𝑖)𝑤𝑖𝑀
𝑗+2
𝑖=𝑗−1
(𝑥)
for the approximation 𝑢(𝑥) when 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1].
Obviously, the relation is valid:
𝑇(𝑥) = ∑ 𝑇(𝑥𝑖)𝑤𝑖𝑀
𝑗+2
𝑖=𝑗−1
(𝑥).
Now we have
|𝑈𝑀(𝑥) − 𝑢(𝑥)| = |𝑈𝑀(𝑥) − 𝑇(𝑥)|
+|𝑇(𝑥) − 𝑢(𝑥)| ≤ 𝐸(𝑢) + | ∑ (𝑇(𝑥𝑖)−𝑢(𝑥𝑖))𝑤𝑖𝑀
𝑗+2
𝑖=𝑗−1
(𝑥)|
≤ 𝐸(𝑢)(1 + 𝜆4𝑀).
The proof is complete.
Let the values of the function 𝑢 be given with errors, i.e.
𝑉(𝑥𝑖) = 𝑢(𝑥𝑖) + 𝜀𝑖 , where |𝜀𝑘| ≤ |𝜀|. In this case, instead of
𝑈(𝑥) = ∑ 𝑢(𝑥𝑖)
𝑗+𝑠
𝑖=𝑗−𝑙+1
𝑤𝑖(𝑥)
we have
𝑉(𝑥) = ∑ (𝑢(𝑥𝑖) + 𝜀𝑖)
𝑗+𝑠
𝑖=𝑗−𝑙+1
𝑤𝑖(𝑥).
Lemma 5. Let 𝑈(𝑥𝑖) = 𝑢(𝑥𝑖) + 𝜀𝑖, where |𝜀𝑖| ≤ |𝜀|. Then
|𝑈(𝑥) − 𝑉(𝑥)|[𝑥𝑗, 𝑥𝑗+1] ≤ 𝜀 𝜆𝑚+1.
Proof. We have
|𝑈(𝑥) − 𝑉(𝑥)|[𝑥𝑗, 𝑥𝑗+1] ≤ ∑ |𝜀𝑖
𝑗+𝑠
𝑖=𝑗−𝑙+1
𝑤𝑖(𝑥)| ≤ 𝜀 𝜆𝑚+1.
The proof is complete.
III. THE CONDENSING GRID OF NODES
Let us now consider the behavior of the Lebesgue constant on
various grids of nodes: condensing according to a certain law
and uniform.
We consider an infinite grid of nodes such that 𝑥𝑗+2 − 𝑥𝑗+1
𝑥𝑗+1 − 𝑥𝑗= 𝑘.
First consider the middle basis splines. We have the
expression for the Lebesgue function for the middle splines:
𝜆4𝑀(𝑥) = |𝑤𝑗−1(𝑥)| + |𝑤𝑗(𝑥)| + |𝑤𝑗+1(𝑥)| + |𝑤𝑗+2(𝑥)|.
Replacing 𝑥, 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], by 𝑥 = 𝑥𝑗 + 𝑡 ℎ, 𝑡 ∈ [0, 1], we
obtain
𝜆4𝑀(𝑥) = −(−2 𝑡 𝑘3 + 2 𝑡2𝑘3 − 𝑘2 − 2 𝑡3𝑘2 + 2 𝑡2𝑘2 − 𝑘
+2𝑡 𝑘 − 4𝑡2𝑘 + 2 𝑡3𝑘 − 2𝑡 + 2 𝑡2)/(𝑘(1 + 𝑘)). The maximum of this expression, when 𝑡 ∈ [0, 1], is reached at
the point
𝑡 = 𝑡0 = −(− 𝑘3 + 2𝑘 − 𝑘2 − 1 + 𝐺𝑀1)/(3𝑘(𝑘 − 1)),
where 𝐺𝑀1 = √𝑘6 − 𝑘5 + 𝑘3 − 𝑘 + 1. The Lebesgue constant can be written in the form:
𝜆4𝑀 = −(4𝑘9 − 6𝑘8 − 12𝑘7 − 6𝑘6𝐺𝑀1 +
13𝑘6 + 6𝑘5𝐺𝑀1 + 3𝑘5 + 3𝑘4-6𝑘3𝐺𝑀1 +
13𝑘3 − 12𝑘2 − 6𝑘 + 6𝑘𝐺𝑀1 + 4 + 2(𝐺𝑀1)3
−6𝐺𝑀1)/(27(1 + 𝑘)(𝑘 − 1)2𝑘3). The graph of 𝜆4
𝑀(𝑥), when 𝑘 various from 1/2 to 2, is shown in
Fig.1.
Fig.1. The plot of the 𝜆4
𝑀(𝑥)
We consider the Lebesque constant for the left basis splines:
𝜆4𝐿(𝑥) = |𝑤𝑗−2(𝑥)| + |𝑤𝑗−1(𝑥)| + |𝑤𝑗(𝑥)| + |𝑤𝑗+1(𝑥)|.
Replacing 𝑥, 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], by 𝑥 = 𝑥𝑗 + 𝑡 ℎ, 𝑡 ∈ [0, 1], we get
𝜆4𝐿(𝑥) = −(2𝑡3𝑘4 − 2𝑡2𝑘4 − 2𝑡 𝑘3 + 2𝑡2𝑘3 − 2𝑡𝑘2 +
2𝑡2𝑘2 − 𝑘 − 1)/(1 + 𝑘). The maximum of this expression, when 𝑡 ∈ [0, 1],, is reached
at the point:
𝑡 = 𝑡0 =𝑘2 − 𝑘 − 1 + √2 𝑘2 + 2𝑘 + 𝑘3 + 1 + 𝑘4
3𝑘2.
The graph of 𝜆4𝐿 , when 𝑘 various from 1/2 to 2, is shown in
Fig.2.
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 443
Fig.2. The plot of the 𝜆4
𝐿
The Lebesgue constant can be written in the form:
𝜆4𝐿 = −(−9𝑘2 − 11𝑘3 + 12𝑘 − 12𝐺𝐿𝑇𝑘2 −
12𝑘𝐺𝐿𝑇 − 6𝑘4𝐺𝐿𝑇 − 6𝑘3𝐺𝐿𝑇 + 2(𝐺𝐿𝑇)3 + 4 −
4𝑘6 − 6𝑘5 − 6𝐺𝐿𝑇)/(27𝑘2(1+k)),
where
𝐺𝐿𝑇 = √2𝑘2 + 2𝑘 + 𝑘3 + 1 + 𝑘4.
Now consider the right basis splines:
𝜆4𝑅(𝑥) = |𝑤𝑗(𝑥)| + |𝑤𝑗+1(𝑥)| + |𝑤𝑗+2(𝑥)| + |𝑤𝑗+3(𝑥)|.
Replacing 𝑥, 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], by 𝑥 = 𝑥𝑗 + 𝑡 ℎ, 𝑡 ∈ [0, 1], we get
𝜆4𝑅(𝑥) = 𝑘4 + 𝑘3 + 2𝑡𝑘2-2𝑡2𝑘2 + 2 𝑡 𝑘 − 2𝑡2𝑘 +
2𝑡3 + 2𝑡 − 4𝑡2)/((1 + 𝑘)𝑘3).
The maximum of this expression, when 𝑡 ∈ [0, 1], is reached
at the point:
𝑡 = 𝑡0 =𝑘2
3+
𝑘
3+
2
3− 𝐺/3,
where 𝐺 = √𝑘4 + 2𝑘3 + 2𝑘2 + 𝑘 + 1. The Lebesgue constant can be written in the form:
𝜆4𝑅 = (−4𝑘6 − 12𝑘5 + 9𝑘4 + 4𝑘4𝐺 + 8𝑘3𝐺 +
11𝑘3 + 8𝑘2𝐺 + 6𝑘 + 4𝑘𝐺 + 4𝐺 + 4)/(27(1 + 𝑘)𝑘3).
The graph of 𝜆4𝑅, when 𝑘 various from 1/2 to 2, is shown in
Fig.3.
Fig.3. The plot of the 𝜆4
𝑅
The approximation error theorems are easy to formulate.
Theorem 1. Let 𝑘 = 1, the grid of nodes is uniform with step
ℎ. Let 𝑢 ∈ 𝐶4([𝑎, 𝑏]). The following statements are valid:
1) |𝑢(𝑥) − 𝑈𝑀(𝑥)| ≤ ℎ4 𝐶𝑀
4!‖𝑢(4)‖
[𝑥𝑗−1,𝑥𝑗+2],
2) |𝑢(𝑥) − 𝑈𝐿(𝑥)| ≤ ℎ4 𝐶𝐿
4!‖𝑢(4)‖
[𝑥𝑗−2,𝑥𝑗+1],
3) |𝑢(𝑥) − 𝑈𝑅(𝑥)| ≤ ℎ4 𝐶𝑅
4!‖𝑢(4)‖
[𝑥𝑗,𝑥𝑗+3],
where 𝐶𝐿 = 𝐶𝑅=1, 𝐶𝑀=0.5625.
Proof. We use the approximation theorem by classical
interpolating polynomials. Let 𝑥𝑗 = 𝑥0 + 𝑗 ℎ, 𝑗 = ±1, ±2, …,
and let 𝑥 ∊ [𝑥𝑗 , 𝑥𝑗+1], be 𝑥 = 𝑥𝑗 + 𝑡 ℎ, 𝑡 ∈ [0, 1]. In order to
estimate the error of approximation we use the formula for the
remainder of the interpolation polynomial (see [17]). In the case
of approximation by the middle basic splines the formula is as
follows:
𝑢 − 𝑈𝑀 =𝑢(4)(𝜉(𝑥))
4!∏ (𝑥 − 𝑥𝑖
𝑗+2𝑖=𝑗−1 ), 𝜉(𝑥) ∊ [𝑥𝑗−1,𝑥𝑗+2].
Thus, in the case of approximation by the middle basic
splines, it is necessary to find the maximum of the polynomial:
𝑆𝑀 = ℎ4 |(𝑡 − 2)𝑡(𝑡2 − 1)|. At the point 𝑡 = 0.5 we have
max𝑡∈[0,1]
𝑆𝑀 = 0.5625 ℎ4 = 𝐶𝑀 ℎ4.
In the case of approximation by the left basic splines, it is
necessary to find the maximum of the polynomial:
𝑆𝐿 = ℎ4 |(𝑡 + 2)𝑡(𝑡2 − 1)|. At the point 𝑡 = 0.6180340 we
have
max𝑡∈[0,1]
𝑆𝐿 = ℎ4 = 𝐶𝐿 ℎ4.
In the case of approximation by the right basic splines, it is
necessary to find the maximum of the polynomial:
𝑆𝑅 = ℎ4 |(𝑡 − 2)(𝑡 − 1)(𝑡 − 3)(𝑡 + 2)𝑡|. At the point 𝑡 =0.3819660 we have max
𝑡∈[0,1]𝑆𝑅 = ℎ4 = 𝐶𝑅 ℎ
4.
The proof is complete.
The next example shows that the constants in the statements
of the Theorem are valid. Let us take ℎ = 0.3, 𝑥0 = 0, 𝑥1 =ℎ, 𝑥2 = 2ℎ, 𝑥3 = 3ℎ.
The actual (Act.err.) and theoretical (Theoret.err.) errors of
the approximation of functions with the right splines on the
interval ℎ = [0, 0.3] are given in the Table 1:
Table 1: The actual (Act.err.) and theoretical (Theoret.err.) errors
of the approximation of functions with the right splines
Function Act. err. Theoret. err.
sin(3𝑥) 0.02297 0.02767
sin(1 + 2𝑥) 0.005111 0.00540
The calculation results given in Table 1 confirm the correctness
of the constants in the approximation estimates of Theorem 1.
IV. THE APPLICATION OF THE LEBESGUE FUNCTIONS
Now we consider a problem. How to select the nodes of the
grid so that when using the approximation with the left splines
on a one grid interval and using the approximation with the
middle splines on the next grid interval then the error of the
approximation error was the same in absolute value.
Consider the approximation with the right splines on one grid
interval and the approximation with the middle splines on the
next grid interval. Solving the equation 𝜆4𝑀(𝑥) = 𝜆4
𝑅(𝑥) we get
𝑘 = 1.380277569. We use the approximation with the right
splines on the grid interval [𝑥𝑗 , 𝑥𝑗+1]. We use the approximation
with the middle splines on the grid interval [𝑥𝑗+1, 𝑥𝑗+2].
Thus, we obtain the relation 𝑥𝑗+2 = 𝑥𝑗+1 + 𝑘(𝑥𝑗+1 − 𝑥𝑗).
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 444
It follows that the grid step in the approximation with the
right splines should be narrower.
The plots of the 𝜆4𝑅 and 𝜆4
𝑀 are given in Fig.4.
Fig.4. The plot of the 𝜆4
𝑅 and 𝜆4𝑀
Consider the approximation with the middle splines on one
grid interval and the approximation with the left splines on the
next grid interval. Solving the equation 𝜆4𝐿=𝜆4
𝑀 we get 𝑘 =0.7245.We use the approximation with the middle splines on
the grid interval [𝑥𝑗 , 𝑥𝑗+1].
We use the approximation with the left splines on the grid
interval [𝑥𝑗+1, 𝑥𝑗+2]. Thus, we obtain the relation
𝑥𝑗+2 = 𝑥𝑗+1 + 𝑘(𝑥𝑗+1 − 𝑥𝑗).
It follows that the grid step in the approximation with the left
splines should be narrower. The plots of the 𝜆4𝐿 and 𝜆4
𝑀 are given
in Fig.5.
Fig.5. The plot of the 𝜆4
𝐿 and 𝜆4𝑀
Consider the approximation with the left splines on one grid
interval and the approximation with the right splines on the next
grid interval. Solving the equation 𝜆4𝑅 = 𝜆4
𝐿 we get 𝑘 = 1.0. We use the approximation with the right splines on the grid
interval [𝑥𝑗 , 𝑥𝑗+1]. We use the approximation with the left
splines on the grid interval [𝑥𝑗+1, 𝑥𝑗+2].
Thus, we obtain the relation
𝑥𝑗+2 = 𝑥𝑗+1 + 𝑘(𝑥𝑗+1 − 𝑥𝑗).
The plots of the 𝜆4𝐿 and 𝜆4
𝑅 are given in Fig.6.
Fig.6. The plot of the 𝜆4
𝑅 and 𝜆4𝐿
V. EXAMPLES
Suppose that the values of the function can be calculated at
an arbitrary point in the interval [𝑎, 𝑏]. How to choose grid
nodes so that on each grid interval the approximation error
would be approximately the same? Near the left end of the
interval [𝑎, 𝑏], we apply the right splines. At the following grid
intervals we can use the middle splines.
Example 1. We approximate the function 𝑢(𝑥) = sin(1 +2𝑥). First we consider the approximation on a uniform grid of
nodes. Let us take ℎ = 0.3, 𝑥0 = 0 , 𝑥1 = ℎ, 𝑥2 = 𝑥1 + ℎ =2ℎ, 𝑥3 = 𝑥2 + ℎ = 3ℎ. The plot of the error of approximation
of the function 𝑢(𝑥) = sin(1 + 2𝑥) with the right splines on
the interval [𝑥0, 𝑥1] and the error of approximation with the
middle splines on the interval [𝑥1, 𝑥2] is given in Fig.7 .
Fig.7. The plot of the error of approximation of the function 𝑢(𝑥) =
sin(1 + 2𝑥) on the uniform grid of nodes (ℎ = 0.3).
Now we move the node 𝑥1 taking into account the ratio 𝑥2−𝑥1
𝑥1−𝑥0=
𝑘, 𝑘 = 1.38, and construct a grid of nodes such that 𝑥0 =0 , 𝑥1 = 0.252, 𝑥2 = 0.6, 𝑥3 = 0.948. On the interval [𝑥0, 𝑥1], we apply the approximation by the right splines, and on the
interval [𝑥1, 𝑥2], we apply the approximation by the middle
splines. Fig. 8 shows that the errors of the approximation in
absolute value with the right and with the middle splines on this
non-uniform grid are almost the same.
Let us construct the special non-uniform grid of nodes. We
take the first interval [𝑥0, 𝑥1] as before. In the first interval, we
apply the approximation by the right splines. In the next
interval, which is [𝑥1, 𝑥2], we apply the approximation by the
middle splines. To calculate the grid node 𝑥2, we use the results
obtained in the previous section.
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 445
Fig.8. The plot of the error of approximation of the function 𝑢(𝑥) =
sin(1 + 2𝑥) on the non-uniform grid of nodes.
Let us take ℎ = 0.3; 𝑘 = 1.38, 𝑥0 = 0; 𝑥1 = ℎ = 0.3; 𝑥2 =𝑥1 + 𝑘(𝑥1 − 𝑥0) = 0.714; 𝑥3 = 𝑥2 + 𝑘(𝑥2 − 𝑥1). The plot of
the error of approximation of the function 𝑢(𝑥) = sin(1 + 2𝑥)
with the right splines on the interval [𝑥0, 𝑥1] and the error of
approximation with the middle splines on the interval [𝑥1, 𝑥2] is given in Fig.9.
For comparison, we give a graph of the approximation error
with a step ℎ = 0.357 (Fig.10).
Fig.9. The plots of the error of approximation of the function 𝑢(𝑥) =sin(1 + 2𝑥) on the non-uniform grid of nodes.
Fig.10. The plot of the error of approximation of the function 𝑢(𝑥) =sin(1 + 2𝑥) on the uniform grid of nodes (ℎ = 0.357).
VI. ABOUT FORMULAS FOR NUMERICAL DIFFERENTIATION
The difference method is widely known and is used both for
solving ordinary differential equations and for solving partial
differential equations. We assume that the solution to the
problem exists and is unique. To construct a difference scheme,
we first need to construct an approximation of the derivatives.
In this section, we will consider the use of non-polynomial
splines to construct formulas for numerical differentiation. Note
that polynomial and non-polynomial splines can be obtained
using the same technique. Since in some cases the
approximation by non-polynomial splines gives a smaller
approximation error, compared to the approximation by
polynomial splines. Therefore, it is hoped that the use of non-
polynomial splines will make it possible to obtain new
computational schemes that give a smaller approximation error
and are computationally stable.
Thus, to construct an algorithm for solving a boundary value
problem with new difference method, we need new numerical
differentiation formulas. In addition, it is necessary to establish
the stability of the obtained scheme. The stability of the
resulting scheme will be discussed later.
The numerical differentiation formulas for the polynomial
case are well known. Nevertheless, we show how to obtain
formulas for numerical differentiation using the formulas of the
middle and left splines.
Differentiating twice the formulas of the middle polynomial
splines we get:
𝑈′′(𝑥𝑗) = 𝑢𝑗−1𝑔′′𝑗−1 + 𝑢𝑗𝑔′′𝑗 + 𝑢𝑗+1𝑔′′𝑗+1 + 𝑢𝑗+2𝑔′′𝑗+2,
where
𝑔′′𝑗−1 = 𝑔′′𝑗+1 =1
ℎ2, 𝑔′′𝑗 = −
2
ℎ2, 𝑔′′
𝑗+2= 0.
Note that in the polynomial case on the uniform set of nodes
the formulas are well known.
On an uneven grid, formulas for the second derivative are used
less frequently. In the polynomial case the formula of the
approximation on the non-uniform grid is the next:
𝑈′′(𝑥𝑗) = 𝑢𝑗−1𝑔′′𝑗−1 + 𝑢𝑗𝑔′′𝑗 + 𝑢𝑗+1𝑔′′𝑗+1 + 𝑢𝑗+2𝑔′′𝑗+2,
where
𝑔′′𝑗−1 = −2(2𝑥𝑗 − 𝑥𝑗+1 − 𝑥𝑗+2)/𝑍𝑛0,
𝑍𝑛0 = (𝑥𝑗 − 𝑥𝑗−1)(𝑥𝑗−1 − 𝑥𝑗+1)(𝑥𝑗−1 − 𝑥𝑗+2),
𝑔′′𝑗 = 2(3𝑥𝑗 − 𝑥𝑗−1 − 𝑥𝑗+1 − 𝑥𝑗+2)/𝑍𝑛,
𝑍𝑛 = (𝑥𝑗 − 𝑥𝑗+1)(𝑥𝑗 − 𝑥𝑗−1)(𝑥𝑗 − 𝑥𝑗+2),
𝑔′′𝑗+1 = 2(2𝑥𝑗 − 𝑥𝑗−1 − 𝑥𝑗+2)/𝑍𝑛1,
𝑍𝑛1 = (𝑥𝑗 − 𝑥𝑗+1)(𝑥𝑗−1 − 𝑥𝑗+1)(𝑥𝑗−1 − 𝑥𝑗+2),
𝑔′′𝑗+2 = −2(2𝑥𝑗 − 𝑥𝑗−1 − 𝑥𝑗+1)/𝑍𝑛2,
𝑍𝑛2 = (𝑥𝑗 − 𝑥𝑗+2)(𝑥𝑗−1 − 𝑥𝑗+2)(𝑥𝑗+1 − 𝑥𝑗+2).
Applying the formulas of the left polynomial splines we get:
𝑈′′(𝑥𝑗) = 𝑢𝑗−1𝑔′′𝑗−1 + 𝑢𝑗𝑔′′𝑗 + 𝑢𝑗+1𝑔′′𝑗+1 + 𝑢𝑗−2𝑔′′𝑗−2,
where
𝑔′′𝑗−1 = 2(2𝑥𝑗 − 𝑥𝑗+1 − 𝑥𝑗−2)/𝑍𝑛0,
𝑍𝑛0 = (𝑥𝑗 − 𝑥𝑗−1)(𝑥𝑗−2 − 𝑥𝑗−1)(𝑥𝑗−1 − 𝑥𝑗+1)
𝑔′′𝑗 = 2(3𝑥𝑗 − 𝑥𝑗−1 − 𝑥𝑗+1 − 𝑥𝑗−2)/𝑍𝑛,
𝑍𝑛 = (𝑥𝑗 − 𝑥𝑗+1)(𝑥𝑗 − 𝑥𝑗−1)(𝑥𝑗 − 𝑥𝑗−2)
𝑔′′𝑗+1 = −2(2𝑥𝑗 − 𝑥𝑗−1 − 𝑥𝑗−2)/𝑍𝑛1,
𝑍𝑛1 = (𝑥𝑗 − 𝑥𝑗+1)(𝑥𝑗−2 − 𝑥𝑗+1)(𝑥𝑗−1 − 𝑥𝑗+1)
𝑔′′𝑗−2 = −2(2𝑥𝑗 − 𝑥𝑗−1 − 𝑥𝑗+1)/𝑍𝑛−2,
𝑍𝑛−2 = (𝑥𝑗 − 𝑥𝑗−2)(𝑥𝑗−2 − 𝑥𝑗−1)(𝑥𝑗−2 − 𝑥𝑗+1).
We construct new formulas based on the previously obtained
non-polynomial splines. Let 𝜑1(𝑥) = 1, 𝜑2(𝑥) = 𝑥, 𝜑3(𝑥) =sin (𝑥), 𝜑4(𝑥) = cos(𝑥). We obtain the formula for the middle
polynomial-trigonometric (non-polynomial) spline (when 𝑥 ∈[𝑥𝑗 , 𝑥𝑗+1]):
𝑈(𝑥) = 𝑢(𝑥𝑗+2)𝑔𝑗+2𝑇 + 𝑢(𝑥𝑗−1)𝑔𝑗−1
𝑇 + 𝑢(𝑥𝑗)𝑔𝑗𝑇(𝑥)
+ 𝑢(𝑥𝑗+1)𝑔𝑗+1𝑇 ),
where
𝑔𝑗𝑇(𝑥) = ((𝑥 − 𝑥𝑗+1)(sin(𝑥𝑗−1 − 𝑥𝑗+2)
+(𝑥𝑗−1 − 𝑥) sin(𝑥𝑗+1 − 𝑥𝑗+2) + (𝑥𝑗+2 − 𝑥) sin(𝑥𝑗−1 − 𝑥𝑗+1)
+(𝑥𝑗+2 − 𝑥𝑗+1) sin(𝑥 − 𝑥𝑗−1) + (𝑥𝑗+1 − 𝑥𝑗−1) sin(𝑥 − 𝑥𝑗+2)
+(𝑥𝑗−1 − 𝑥𝑗+2) sin(𝑥 − 𝑥𝑗+1))/Z,
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 446
where
𝑍 = (𝑥𝑗 − 𝑥𝑗+1) sin(𝑥𝑗−1 − 𝑥𝑗+2) +
(𝑥𝑗−1 − 𝑥𝑗) sin(𝑥𝑗+1 − 𝑥𝑗+2) +
(𝑥𝑗+2 − 𝑥𝑗) sin(𝑥𝑗−1 − 𝑥𝑗+1) +
(𝑥𝑗+2 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗−1) +
(𝑥𝑗+1 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗+2) +
(𝑥𝑗−1 − 𝑥𝑗+2),
𝑔𝑗+1𝑇 (𝑥) = ((𝑥𝑗 − 𝑥) sin(𝑥𝑗−1 − 𝑥𝑗+2) +
(𝑥𝑗+2 − 𝑥) sin(𝑥𝑗 − 𝑥𝑗−1)+(𝑥 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗+2)
+(𝑥𝑗 − 𝑥𝑗+2) sin(𝑥 − 𝑥𝑗−1) + (𝑥𝑗−1 − 𝑥𝑗) sin(𝑥 − 𝑥𝑗+2)
+(𝑥𝑗+2 − 𝑥𝑗−1) sin(𝑥 − 𝑥𝑗))/𝑍,
𝑔𝑗+2𝑇 (𝑥) = ((𝑥 − 𝑥𝑗) sin(𝑥𝑗−1 − 𝑥𝑗+1) +
(𝑥 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗−1) + (𝑥𝑗−1 − 𝑥) sin(𝑥𝑗 − 𝑥𝑗+1)+
(𝑥𝑗+1 − 𝑥𝑗) sin(𝑥 − 𝑥𝑗−1) + (𝑥𝑗−1 − 𝑥𝑗+1) sin(𝑥 − 𝑥𝑗) +
(𝑥𝑗 − 𝑥𝑗−1) sin(𝑥 − 𝑥𝑗+1))/𝑍,
𝑔𝑗−1𝑇 (𝑥) = ((𝑥 − 𝑥𝑗) sin(𝑥𝑗+1 − 𝑥𝑗+2) +
(𝑥 − 𝑥𝑗+2) sin(𝑥𝑗 − 𝑥𝑗+1) + (𝑥𝑗+1 − 𝑥) sin(𝑥𝑗 − 𝑥𝑗+2) +
(𝑥𝑗 − 𝑥𝑗+1) sin(𝑥 − 𝑥𝑗+2) + (𝑥𝑗+2 − 𝑥𝑗) sin(𝑥 − 𝑥𝑗+1) +
(𝑥𝑗+1 − 𝑥𝑗+2) sin(𝑥 − 𝑥𝑗)/𝑍.
Based on the middle basis polynomial-trigonometric splines,
we obtain the formula:
𝑈′′(𝑥𝑗) = 𝑈(𝑥𝑗−1)𝑔′′𝑗−1 + 𝑈(𝑥𝑗)𝑔′′𝑗 + 𝑈(𝑥𝑗+1)𝑔′′𝑗+1
+ 𝑈(𝑥𝑗+2)𝑔′′𝑗+2),
where
𝑔′′𝑗 = ((𝑥𝑗+1 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗+2) +
(𝑥𝑗−1 − 𝑥𝑗+2) sin(𝑥𝑗 − 𝑥𝑗+1) +
(𝑥𝑗+2 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗−1))/𝑍𝑧,
𝑔′′𝑗+1 = ((𝑥𝑗 − 𝑥𝑗+2) sin(𝑥𝑗 − 𝑥𝑗−1) +
(𝑥𝑗−1 − 𝑥𝑗) sin(𝑥𝑗 − 𝑥𝑗+2))/𝑍𝑧,
𝑍𝑧 = (𝑥𝑗+1 − 𝑥𝑗) sin(𝑥𝑗−1 − 𝑥𝑗+2) +
(𝑥𝑗 − 𝑥𝑗−1) sin(𝑥𝑗+1 − 𝑥𝑗+2) +
(𝑥𝑗 − 𝑥𝑗+2) sin(𝑥𝑗−1 − 𝑥𝑗+1) +
(𝑥𝑗+1 − 𝑥𝑗+2) sin(𝑥𝑗 − 𝑥𝑗−1) +
(𝑥𝑗−1 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗+2) +
(𝑥𝑗+2 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗+1),
𝑔′′𝑗+2 = ((𝑥𝑗+1 − 𝑥𝑗) sin(𝑥𝑗 − 𝑥𝑗−1) +
(𝑥𝑗 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗+1))/𝑍𝑧,
𝑔′′𝑗−1 = ((𝑥𝑗 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗+2) +
(𝑥𝑗+2 − 𝑥𝑗) sin(𝑥𝑗 − 𝑥𝑗+1))/𝑍𝑧.
When ℎ = 𝑐𝑜𝑛𝑠𝑡 we have the formula for the approximating
the second derivative of the function:
𝑈′′(𝑥𝑗) = 𝑢𝑗−1𝑔′′𝑗−1 + 𝑢𝑗𝑔𝑗′′ + 𝑢𝑗+1𝑔′′𝑗+1 + 𝑢𝑗+2𝑔′′𝑗+2,
where
𝑔′′𝑗−1 =2 sin(ℎ) − sin (2ℎ)
sin(3ℎ) + 5 sin(ℎ) − 4sin (2ℎ),
𝑔′′𝑗 =2 sin(2ℎ) − 4sin (2ℎ)
sin(3ℎ) + 5 sin(ℎ) − 4sin (2ℎ),
𝑔′′𝑗+1 =2 sin(ℎ) − sin (2ℎ)
sin(3ℎ) + 5 sin(ℎ) − 4sin (2ℎ).
𝑔′′𝑗+2 = 0.
Based on the left basis splines, we obtain the formula for the
second derivative:
𝑈′′(𝑥𝑗) = (𝑈(𝑥𝑗−2)𝑔′′𝑗−2 + 𝑈(𝑥𝑗−1)𝑔′′𝑗−1 + 𝑈(𝑥𝑗)𝑔′′𝑗
+ 𝑈(𝑥𝑗+1)𝑔′′𝑗+1)/𝑍𝑛,
where
𝑔′′𝑗−1 = (𝑥𝑗 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗−2) +
(𝑥𝑗−2 − 𝑥𝑗) sin(𝑥𝑗 − 𝑥𝑗+1),
𝑔′′𝑗−2 = (𝑥𝑗+1 − 𝑥𝑗) sin(𝑥𝑗 − 𝑥𝑗−1) +
(𝑥𝑗 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗+1),
𝑔′′𝑗 = (𝑥𝑗−2 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗−1) +
(𝑥𝑗+1 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗−2) +
(𝑥𝑗−1 − 𝑥𝑗−2) sin(𝑥𝑗 − 𝑥𝑗+1),
𝑔′′𝑗+1 = (𝑥𝑗−1 − 𝑥𝑗) sin(𝑥𝑗 − 𝑥𝑗−2) +
(𝑥𝑗 − 𝑥𝑗−2) sin(𝑥𝑗 − 𝑥𝑗−1),
𝑍𝑛 = (𝑥𝑗 − 𝑥𝑗+1)sin (𝑥𝑗−2 − 𝑥𝑗−1)+
(𝑥𝑗−1 − 𝑥𝑗) sin(𝑥𝑗−2 − 𝑥𝑗+1) + (𝑥𝑗 − 𝑥𝑗−2) sin(𝑥𝑗−1 − 𝑥𝑗+1) +
(𝑥𝑗−1 − 𝑥𝑗+1) sin(𝑥𝑗 − 𝑥𝑗−2) + (𝑥𝑗−2 − 𝑥𝑗−1) sin(𝑥𝑗 − 𝑥𝑗+1) +
(𝑥𝑗+1 − 𝑥𝑗−2) sin(𝑥𝑗 − 𝑥𝑗−1).
It can be obtained if the grid of nodes is uniform, then the
relation is valid: 𝑢(𝑥𝑗) = 𝑈(𝑥𝑗) + 𝑂(ℎ2).
Nevertheless, we show how to obtain formulas for numerical
differentiation using the formulas of the middle and left splines.
On the non-uniform set of nodes the formulas for the first
derivative such, that 𝑢′(𝑥𝑗) = 𝑈′(𝑥𝑗) + 𝑂(ℎ2), can be written
in the form:
𝑈′(𝑥𝑗) = (𝑢(𝑥𝑗−1)𝑔′𝑗−1 + 𝑢(𝑥𝑗)𝑔′𝑗 + 𝑢(𝑥𝑗+1)𝑔′𝑗+1)/𝑍𝑛,
where
𝑔′𝑗−1 = (cos(𝑥𝑗 − 𝑥𝑗+1) − 1)/2,
𝑔′𝑗+1 = (1 − cos(𝑥𝑗 − 𝑥𝑗−1))/2,
𝑔′𝑗 = (cos(𝑥𝑗 − 𝑥𝑗−1) − cos(𝑥𝑗 − 𝑥𝑗+1))/2,
𝑍𝑛 = 2 sin(𝑥𝑗/2 − 𝑥𝑗−1/2) sin(𝑥𝑗−1/2 − 𝑥𝑗+1/2) sin(𝑥𝑗/2 −
𝑥𝑗+1/2).
On the uniform set of nodes the formula has the form:
𝑢′(𝑥𝑗) = (𝑢𝑗+1𝑔′𝑗+1 − 𝑢𝑗−1𝑔′𝑗−1)/(2 sin (ℎ)) + 𝑂(ℎ2).
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 447
In the polynomial case we have:
𝑈′(𝑥𝑗) = (𝑢(𝑥𝑗−1)𝑔′𝑗−1 + 𝑢(𝑥𝑗)𝑔′𝑗 + 𝑢(𝑥𝑗+1)𝑔′𝑗+1)/𝑍𝑛,
where
𝑔′𝑗−1 = −(𝑥𝑗 − 𝑥𝑗+1)2
, 𝑔′𝑗+1 = (𝑥𝑗 − 𝑥𝑗−1)2
,
𝑔′𝑗 = (2𝑥𝑗 − 𝑥𝑗+1 − 𝑥𝑗−1)(𝑥𝑗−1 − 𝑥𝑗+1),
𝑍𝑛 = (𝑥𝑗 − 𝑥𝑗−1)(𝑥𝑗 − 𝑥𝑗+1)(𝑥𝑗−1 − 𝑥𝑗+1).
On the uniform set of nodes we have:
𝑈′(𝑥𝑗) = (𝑢(𝑥𝑗+1) − 𝑢(𝑥𝑗−1))/(2ℎ).
We apply the obtained formulas to solve the boundary value
problem by the difference method (mesh method).
Example 2. Let us have a problem:
𝑢′′ + 𝑝(𝑥)𝑢′ − 𝑢 = 𝑓(𝑥), 0 ≤ 𝑥 ≤ 1, 𝑢(0) = 𝑢(1) = 0, 𝑝(𝑥) = sin2(1 − 𝑥).
We constructed the right side of the equation using the solution
of this equation (i.e., the function 𝑢(𝑥) = sin(𝑥) sin(𝑥 − 1) ). Thus, we have:
𝑓(𝑥) =5
2cos(2𝑥 − 1) +
1
2sin(2𝑥 − 1) −
1
4sin(4𝑥 − 3)
−cos(1)
2−
sin (1)
4.
The minimum number of internal nodes for using our splines
should be 3. Table 2 shows the results of applying numerical
differentiation formulas constructed using polynomial and non-
polynomial splines. It shows the actual errors of the
approximate solutions at the grid nodes with the step ℎ = 0.2,
constructed using polynomial and non-polynomial splines. The
example shows that in the case of a trigonometric solution, the
use of non-polynomial splines takes precedence.
Table 2. The actual errors of the approximate solution obtained using
polynomial or non-polynomial splines on a uniform grid of nodes.
ℎ = 0.2 Pol.spl. Non-pol.spl.
𝑥1 = ℎ -0.00160 -0.00120
𝑥2 = 2ℎ -0.00256 -0.00191
𝑥3 = 3ℎ -0.00261 -0.00196
𝑥4 = 4ℎ -0.00172 -0.00129
Let us denote by 𝑈 the approximate solution that we obtain as
a result of solving the problem. The plots of the errors of the
approximate solution in absolute value when 𝑛 = 10 are given
in Figures 11-15. The plot of the errors in absolute value of the
approximate solution constructed using non-polynomial splines
on the uniform grid of nodes with step ℎ = 0.1 is given in the
Figure 11.
Fig.11. The plot of the errors in absolute values of the approximate
solution constructed on the uniform grid of nodes when 𝑛 = 10
Fig.12. The plot of the actual errors in absolute values of the
approximate solution constructed using non-polynomial splines on
the uniform grid of nodes when 𝑛 = 10
The plot of the errors in absolute value of the approximate
solution constructed using polynomial splines on the uniform
grid of nodes with step ℎ = 0.1 is given in Figure 12. Note that
in both cases it is easy to the connect points, which were
obtained with the difference method, with a line that was
constructed by using the formulas of the corresponding spline
approximations. Figure 15 shows the points obtained with the
difference method and the line which connects the points. We
also call the line, which we constructed with the polynomial or
non-polynomial splines with the fourth order of approximation,
the approximate solution. Figure 12 shows the errors in absolute
value of the approximate solution when we constructed the
solution between the grid nodes using the formulas of non-
polynomial splines. Figure 14 shows the errors in absolute
value of the approximate solution when we constructed the
solution between the grid nodes using the formulas of the cubic
polynomial splines. Thus, the use of a special non-uniform grid
allows us to reduce the error in constructing the solution of the
boundary value problem.
Fig.13. The plot of the errors in absolute values of the approximate
solution constructed on the uniform grid of nodes when 𝑛 = 10
Now consider an equation with full coefficients. we will solve
it first using the traditional method, then using trigonometric
approximations.
Fig.14. The plot of the actual errors in absolute values of the
approximate solution constructed using polynomial splines on the
uniform grid of nodes when 𝑛 = 10
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 448
Fig.15. The plot of the approximate solution (red points) and
approximate solution constructed using polynomial splines on the
uniform grid of nodes when 𝑛 = 10
Example 3. Let us have a problem:
𝑢′′ + 𝑝(𝑥)𝑢′ − 𝑢 = 𝑓(𝑥), 0 ≤ 𝑥 ≤ 1, 𝑢(0) = 𝑢(1) = 0, 𝑝(𝑥) = (1 − 𝑥)2.
We constructed the right side of the equation using the solution
of this equation (i.e., the function 𝑢(𝑥) =(𝑥)2 (1 − 𝑥)2). Thus,
we have: 𝑓(𝑥) = 2 − 10𝑥 + 𝑥2 + 20𝑥3 − 15𝑥4 + 4𝑥5.
First, we construct a solution using the usual difference method
on a grid of nodes with a step of 0.1. The error of the solution
obtained by the usual difference method (polynomial splines) at
the nodes of a uniform grid is presented in the second column
of Table 3. The error of the solution obtained using the non-
polynomial splines approximation at the nodes of a uniform
grid is presented in the third column of the Table 3. Figure 16
shows a graph of the solution error (in absolute value), obtained
using non-polynomial trigonometric splines.
Table 3. The errors of the solution obtained by the usual difference
method (Pol.spl.) and using the non-polynomial splines
(Non-pol.spl.)
ℎ = 0.1 Pol.spl. Non-pol.spl.
𝑥1 = ℎ -0.000711 -0.000707
𝑥2 = 2ℎ -0.00131 -0.00129
𝑥3 = 3ℎ -0.00176 -0.00173
𝑥4 = 4ℎ -0.00205 -0.00201
𝑥5 = 5ℎ -0.00216 -0.00212
𝑥6 = 6ℎ -0.00210 -0.00206
𝑥7 = 7ℎ -0.00186 -0.00183
𝑥8 = 8ℎ -0.00143 -0.00141
𝑥9 = 9ℎ -0.000811 -0.000805
Fig.16. The plot of the errors in absolute values of the approximate
solution constructed on the uniform grid of nodes when 𝑛 = 10
Examples 2 and 3 show that the use of trigonometric splines to
approximate derivatives can improve the error of the resulting
solution to the boundary value problem. In the future, it is
supposed to consider in more detail the use of other non-
polynomial splines for solving boundary value problems.
VII. ABOUT STABILITY
The Sturm-Liouville problem 𝐿𝑢 = 𝑟(𝑥), 𝑎 ≤ 𝑥 ≤ 𝑏, where
𝐿𝑦 = −𝑦′′ + 𝑝(𝑥)𝑦′ + 𝑞(𝑥)𝑦, with the simplest boundary
conditions 𝑦(𝑎) = 𝛼, 𝑦(𝑏) = 𝛽, was investigated in [17]. In
particular, the author of this book noted that if 𝑝, 𝑞, 𝑟 are
continuous and 𝑞 positive on [𝑎, 𝑏] then the problem has a
unique solution. Suppose that there are positive constants 𝑝, 𝑞,
𝑞 such that |𝑝(𝑥)| ≤ 𝑝,̅ 0 < 𝑞 ≤ 𝑞(𝑥) ≤ �̅� for 𝑎 ≤ 𝑥 ≤ 𝑏.
A simple finite difference operator, acting on a grid function u
which approximates the operator 𝐿 is
𝐿ℎ = −𝑢𝑗+1 − 2𝑢𝑗 + 𝑢𝑗−1
ℎ2+ 𝑝(𝑥𝑗)
𝑢𝑗+1 − 𝑢𝑗−1
2ℎ+ 𝑞(𝑥𝑗)𝑢𝑗,
𝑗 = 1,2, … , 𝑁. Theorem 7.3.1 (see [17]) states that if ℎ𝑝 ≤ 2, then 𝐿ℎ is stable.
For non-polynomial finite-difference operators considered in
the previous section, a similar statement holds. The proof is
similar to that given in the book.
VIII. CONCLUSION
When approximating by the splines of the maximum defect
on a finite grid of nodes, it is necessary to use the left, the right,
and the middle splines. The right splines are applied near the
left end of the interval [𝑎, 𝑏], The left splines are applied near
the right end of the interval [𝑎, 𝑏]. The left and the right splines
give a large error in absolute value, compared with the middle
splines, in the estimation of the approximation error.
Changing the length of the grid intervals, we can make the
approximation error in absolute value when using different
splines be approximately the same.
In this work, constants are found that make it possible to use
the left, the right, or the middle splines at different grid
intervals, so that the approximation error in absolute value is
the same. The application of information on the behavior of the
Lebesgue constants allows us to construct a grid of nodes that
allows us to improve the quality of approximation.
In the future, similar theorems will be considered for
approximation with the trigonometric splines [16], and the
numerical experiment will be discussed in detail. The left, the
right and the middle local cubic polynomial splines are
conveniently used for approximating functions, including for
visualizing the solution of a differential equation. Non-
polynomial local splines of the fourth order of approximation
are also conveniently used for approximating functions,
including for visualizing the solution of a differential equation.
In addition, they allow us to obtain formulas for numerical
differentiation with new properties, which can be useful in the
construction of difference schemes for solving partial
differential equations. This will be the subject of further
research.
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 449
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This article is published under the terms of the Creative Commons Attribution License 4.0 https://creativecommons.org/licenses/by/4.0/deed.en_US
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2020.14.59 Volume 14, 2020
ISSN: 1998-4464 450
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